"Indivisibillity of goods" is a standard example of a non-differentiable feasible set. Still, while it has produced a number of theoretical results in microeconomics mainly regarding individual behavior, when examining real-world markets and economies, the smoothing effects of aggregation allows to treat it as though it was smooth and differentiable, with the approximation error being negligible (and indeed it is).
An interesting case that might qualify for what you are asking, is dynamic problems where investment becomes a step-function (see this post also).
A classic example is Telecommunications market. Companies invest in building an initial network that has a capacity "to last certain periods". As (or if) they grow commercially, it comes a time that capacity is indeed reached (typically 80% of theoretical capacity, validating once more the Pareto principle/rule-of-thumb), and then they have to invest, not a little something to smoothly increase capacity say by 1%, but again a sizeable amount to increase capacity to "last for some periods". Etc.
This sometimes is inherent in the nature of things and the technology involved, and/or takes into account economies of scale from volume-purchases and project-management costs.
What this does is to impose an additional dynamic constraint on the decision variable "investment": "if network saturation is below XX, Investment is "zero", if it has reached XX, invest, and not less than YY". So in the first subset of periods, the feasible set for investment is just a single point (zero), while in the rest, it has a non-trivial lower bound. In turn the "network saturation" will depend on other decision variables of the firm (like marketing efforts etc) as well as on past investments that have determined the current maximum capacity.
Again, at the macroeconomic level, one could invoke the smoothing effects of aggregation, but not when doing micro-economic work on how to solve the problem of a particular firm. Obviously, this has specific applied uses.